Discussions

0:00 It's just this business of using the classifying rings in the various algebraic theories to show how they intersected, so to speak. Is this going to be okay with you, the Highbury Vaults, to have lunch? Or would you prefer to go across the road to where we were? This is the pub where we were the first night. I think you were the first night. Okay. It's good for me. I like it. It's a nice old pub. Thanks. Hello. Hi. Are we okay for lunch? Ah, this is it. I should have read that long ago. Yes, that's fatal whenever anybody tries to sort through books and papers. You get many interesting things come to the surface. Not to speak of letter B, it's even worse. I certainly know that feeling, maybe too well. The last two times I've had to move house, I mean, it was just dreadful. I ended up chucking three quarters of the stuff into a skip because, of course, when I'd started sorting through it, that's exactly what had happened. Speaking of going through old papers, I was going through, I found, pretty near the top of the pile, a couple of your papers with Gonzalo on the fractional exponents, and I realized something I hadn't clearly realized before, that you have a preprint type of paper, which, which, He talks about second-order differential equations and the fact that they form a topos and related matter based on some very interesting two-dimensional calculations, two much better than the ones that I did. And then you have the formal publication, which as far as I can tell only has the general categorical stuff and doesn't deal with the fact that it's about second-order equations at all, or not mistaken. So in other words, this latter part has never been formally published. There's only my informal thing that you got in his thing.

2:30 I don't even think that we have a pre-publication because... Yeah, there wasn't 98 or 99. 98 must have been. Because I commented on it in my email, in my internet website, it still exists. I commented on that informal paper and it was November of 98, I think. No, but I was just assuming that since you did have a formal publication, that this material was available in some formal publication. Now I suddenly got the impression that it didn't. Maybe you intended to have a second form of this, I don't know. Hmm, 98, but then it should be on the archive or something. I mean, we wouldn't write anything. In a semi-publication form without putting it on the archive, it may be. I think it's probably listed on your home page, which refers to the archive. Okay, so it would be accessible. So the fact that the second order of equations can be construed to form a tuple, which... Well, the categorical preliminaries were published in this TAC article. Oh, yeah, that's right. Yeah, that's right. That's pure category theory, so to speak. Yeah. Okay. It's a warehouse. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. And so in particular, you see, I think the archives have not reviewed them at the end. They never have had the opportunity to deal with this subject. Of course, they often refrain from dealing at all, even when they haven't. Anyway, this is just a remark about the present state of affairs. It should, of course, publish more. The book should publish more, either separately or together. You still have to get around to putting quite a number of your older papers on your site, don't you? You've done a tremendous amount in the last few years, and writing new introductions to them, like you did with Colin for the ETCS. Yeah, yeah, the commentaries. Yeah, the commentaries, those are absolutely wonderful, and indeed the commentary on your

5:00 thesis. I hope you're going to get around to writing a commentary and to a new... The publication of the of the Eilenberg-Feschrift paper, the Eilenberg-Feschrift paper, you know, you know the one I mean, well, 76, I think it actually appeared, 75 as they, you probably wrote it in 74, yeah. The Eilenberg Festschrift, there was a Festschrift for the 60th, I think it was his 60th, and there was a Festschrift which came out by Miles Tierney, which came out a year later, 75, and Bill Husserl. There's a wonderful paper in there, which was the first of your papers that I really tried to study seriously and got a great deal out of, and even now there's still so many. That's one of the things that I recall very clearly from it, one of the many, many wonderful insights in it, I remember. Yes, but I think it would be very well worth revisiting and writing an introduction to that. Variable structures and variable quantities in topology is the title, I think, but I just think, anyway, that would be very well worth it. Making one of the re-publications. And the subject of what happens to people's... In 73 there was a meeting here in Bristol. Oh, you published that. That's another one. I think that's the next one you should also re-publish with an introduction. The Bristol paper. Continuously variable... Yes, that's right. Geometric logic. Geometric logic equals algebraic geometry. Yeah, that's right. Deleu, it's downstairs. No, no, no. Sorry. Yes, you can get coffee. You can get coffee in a pub. Yes, these days the British are getting quite civilized. I had a nice big bowl of coffee, like in Buffalo, for breakfast this morning. They will give you a proper coffee, I think. Oh, you just want a little one? But now I want a small one. Okay, well, we'll just go and ask them. If you already ordered coffee, you may order one. Yeah, I'll pay. I'll order one. I'll have one as well. I'd actually prefer, oh yes, actually an espresso at the moment, that would be good, but I need it to stay, no, they probably won't be able to do espresso, you know, in a pub, just for coffees, they'll just come up with instant coffee, I'm afraid, but still it'll be coffee, what the hell.

7:30 No, that's right, the 1973 Bristol Colloquium paper. If I understood correctly, John Mayberry attended that meeting, I don't remember him from that time. The first time I remember Mayberry is when you were in Cambridge. I don't know if he attended that or not. He was certainly here in Bristol at that time. Yes, he must have done, because it was a logic colloquium meeting. Yes, I guess he probably didn't register. Could be. I mean, I've never asked him about that. He must have. He must have done. Unless for some reason he was having a sabbatical that year or something, I don't know. Well, they are absolutely wonderful papers, the two, both of them. In fact, I read them in the opposite order. I read the later one. Variable structures and variable quantities in topi first, and then the one that you've published in the Bristol Logic Colloquium Proceedings, but in both cases I was completely, it was literally the business about scales falling from your eyes and all that stuff that Kant says and being awoken from dogmatic slumber. Yeah, those two are certainly crucial stages in the development of the whole point of view. They were certainly crucial stages in my life, I have to say. Together they certainly changed the whole course of my life, there's no question about that. For that alone I would be entirely grateful to you, if it wasn't for all the other things I have reason to be grateful for. But on the subject of what happens to people's papers if they're not careful, I heard a terrible story from Steve Aldi when we were in Oberwolfach in February about what happened to Saunders's papers and Saunders's... Did you hear about this after he died? Well, it was terrible, his... Daughter, this is what Steve says, I don't know how far he's to be trusted, but he was McLean's assistant and, you know, he's very close to him. He said that he had arranged, after Saunders died, he had arranged to come down from, you know, where he is in Pittsburgh these days, to take charge of Saunders' papers and his networks and put in order.

10:00 Saunders' daughter had rung up the math department in Chicago and asked them if they were going to come and clear out his papers. These were the papers that he kept in the house, not in his office I should say. But of course he kept a lot of papers, important material in the house. And the guy who is the head of the department there now, who... Aldi says it's just a complete joke. I don't know who he is. Not Peter May. Peter May is successful? I don't know the name. I can't remember the name. But he says he's a complete jerk. He says he's a scrumptious whippersnapper. No appreciation whatever of what Saunders did or what his life's work represents for science, for mathematics, just said more or less that we're not interested. You know, do what you like with them. So she just chucked the whole lot. When Steve arrived, she had chucked out half of it onto a skip and the rest was just lying on the lawn. His story, I mean, he went through, he rescued, but there were books there from Saunders' library which had detailed notations, many, many, many marginal notations which were obviously of extraordinary value to any serious historical mathematics, which he had to rescue from a skip. Because the people of Chicago would not even prepare to take responsibility for that. I think that's the most dreadful story. Most of his papers, as Steve confirmed, most of his papers he did put in order at the time of his retirement and he, you know, he left in the keeping of the university, but there was a lot of material which he had at home at the time that he died which I was, according to Steve, was thrown out. I don't, I can't, I can only just report what he, he said, and he felt very, you know, a bit of regret that he hadn't gone straight down and, you know, tended to the matter right away, but he never believed that they would, you know,

12:30 Well, there are a lot of these practical stories, a lot of these criminal stories. Yes, as I say, I don't know. I only just report what he said. That would do all the records. Especially the incriminating stuff. Yes, of course. This is why we have difficulty in writing Roman history. Thank you. Oh yes. Who did it and when? Now, of course, it's an even more for historians. Yeah. For modern historians, this is, you know, almost half the, this is the, this is the issue which takes up almost half of their time and energy. Precisely. You only have to look at... Just to take an example at random, Ciano's diaries. Ciano, the son-in-law of Mussolini, the foreign minister of Mussolini's fascist regime, who was also his son-in-law. Yeah, a guy who was actually executed on Mussolini's orders because he was one of the people who voted against him when he was thrown out of power before the... Ciano had kept a diary for 20 years. He kept extremely detailed diaries of his period as foreign minister, of his conversations with the Nazis and with Mussolini and other people. And these contained a great deal of material which Mussolini obviously didn't want to see the light of day. And he managed to smuggle them out to Switzerland. One of his daughters smuggled them out to Switzerland. And his wife attempted to use them, in fact, as leverage to get Mussolini to spare his life. It didn't work. In fact, Mussolini wanted to spare his life, largely because he was his son-in-law and his daughter was frantic, but Hitler, of course, insisted that they must all be executed, so they were. And the duars did eventually surface after the war. The British, in fact, got hold of them. And by means of which is still extremely unclear. It's pretty obvious the British Secret Service was involved. They came into the hands of a British journalist, quite influential, very anti-Soviet British journalist, the editor of the Daily Telegraph, a man called Malcolm Muggeridge. And he edited them for publication. It's obvious that they were very, very...

15:00 Nobody has ever seen the actual manuscript version. It's obvious that they were, well, I said obviously some people have seen it, but they have never been, they have never been made public, and it's obvious, just through internal evidence, that they were very heavily weeded, and particularly everything that Ciano says about, for instance, secret negotiations with the British and with Churchill, of course, was a great admonishment. They had quite a long personal correspondence, which again has disappeared. All of that was very carefully weeded and taken out. It's a sort of difficult paper, but nobody has read it seriously, I believe, and you know the paper by Bollner and Dubuc on German representability. Ah, I see. Yeah, so that's one of the things on my agenda. This I should read and understand. I also had some correspondence with Marta recently on it. So that's some interesting stuff in it but I mean it's the constantly they have a This notion of an open sub-object of an object, any topos in fact, a purely logical notion of what they call the non-open, which is crucial for their theory of germs, you cannot have a good notion of germ unless you have a good notion of open, and they do it in order to have... To be able to formulate, just to have a synthetic formulation of something like Inverse Function Theorem, which says that if it's infinitesimally invertible, it's germ-invertible, I'd say.

17:30 Is that implication of the theorem? Yeah, that, well... Then there's something else, again. That's something else, but just to have... So to speak, the power series converges. That's one of the, so to speak, classical combinations. But anyway, if a function from Rn to Rn has invertible one jet at zero, then it has an invertible germ at zero. That's the inverse function theory. And the theory is so much... Well, part of the point is to formulate what does germ mean. So one of the achievements of the paper is that I can just formulate it. And I believe even the proof in some models that it holds, and also some things about local solutions of first-order differential equations, existence and uniqueness, that also doesn't exist globally, so that again is really a germ existence. And so they did something in that table which I think is important, but then they both left at least not on this issue. Well, you know, Oscar Bruno was a student. I've never met him. Anyway. There's an indication of how they all understand each other. I thought that he had really settled the question of what open sets and constrictors are and so forth and so on, because he considers, you know, the contrast between open, per se, and exponentialism, and then he looked completely at, you know, Whitney topology, but he's not that prepared. But, oh great, he's got it now.

20:00 Then when I finally met him, he said, he said in essence, well, really the essence of all this, as I showed, by non-opening the book, that kind of a statement doesn't, one doesn't make in the actual paper, so I didn't even realize that all this interesting work, as far as the original problem was, that the result should be far less negative, primarily. But, yeah, there's something strange about these Penrose things. It's hard to explain. On the one hand, it's very amazing how it actually agrees with the, quote, usual notion of who are the representable functions. But then the question is, since it's a purely logical notion, it makes sense for arbitrary objects, and it should be somehow sensible for function spaces as well, but it just seems that when you apply it to function spaces, it's a congenial, and I'm not agreeing with this classical notion at all. Well, there's a more basic logical objection to it. I mean, it's not representable. You'd think that there's something like the Sierpinski space, in other words, a sub-object of omega, which would just represent the n's. I think an algebraic geometry is such a thing. Mainly, if you take the line, r... And then, invertible elements in R. It's a subspace, but it has a classifying object, omega, and if you take the image of that, if you take the image of that, that will be an object that classifies, well, not general opens, but locally closed. I mean, that represents something, but that's only because of... These problems, I spend a lot of time and energy to, because in the book I've been writing, I need the notion of open. Not very seriously, but...

22:30 Just in order to see what is a manifold, this depends on, there's an open copper, but I decided to take a sort of an axiomatic. It's a class of monic maps stable on certain constructions. Essentially, the Sharia are a little more like axiomatics. So it's an external given. It's not something you can define. And hence, also, and that took me some time. Hence, not something where you can make chief semantics. You cannot say, you cannot make the assertion this object, well, a specific global object, you could say this is a manifold, because if you can find an external given open cover, but it doesn't make sense to say for all... Subspaces of a space that come about in a given way must be a manifold. For instance, if you want to say the fibers of a submersion are manifolds. This is a statement about a family of subspaces which are not just permatized by points but by general elements. And hence you have to have the... You need a more internal notion to open that up. Yeah, before it has an interpretation in chief semantics. So I can say this Grassmann manifold, which I have constructed here, is a manifold. That's alright. But I cannot say the fibers of the submersion are always manifolds. Because the notion of manifold doesn't make sense in the internal language. With, when I take the notion of open as something externally given. So that's not a too heavy situation, but it's unworkable if you take a manifold to be, we're not talking about splitting from the important to the hard to the end. Yeah, I think that would work. No, but it's just putting it in an open set. That makes the solution very different.

25:00 You see. Maybe the input part is fine, but what is an open set of Rn? And can you quantify the relief? Yeah, you cannot quantify over the family of open sets of something. There's not a space of open subspaces of a space. But on the other hand, for Aryans, maybe not the general Aryans, the Newton subsets is just given by a math tool. There's just a smooth function. The place is where it's invertible. You take the invertible elements of r, which is the same as the invertible elements of d to the d. So, in other words, you think of the automorphisms of D, which is included in D to the D, as sort of a generic open set, and now any pullback of that should be open, actually more general things should be open, but at least that, and that's good for Rn, as long as it's Rn, you can get it to open that way. I feel the moment with me. Every open subset of Rn. All of this has a characteristic function, a smooth function to R such that those points that hit non-single, i.e. invertible, form the subspace. So in that sense... So that's a viable notion and then you can take the input... And then you get something as classic as it would be exactly with the metaphors, by using two or three different non-trivial theorems and putting them together. But, you see, so, I must say, this is just the one reason I never liked the whole idea of well-adapted. See, starting with the data, which is ordinary manifolds, antitropos, and an embedding with all these properties, the real thing is, well, you can intrinsically define what these manifolds should be, and for one thing, this will be a more general theory because of the other examples. But moreover, at any point you can hypothesize that this is as much like the classical mathematics.

27:30 You like, but you don't need that extra structure floating around. So I really don't like this. No, because it makes differential geometry subordinate to the infinity geometry. But differential geometry is more basic. Yeah, and he's also of course, you know, yeah, I mean, and it's totally, yes, and obviously it rests entirely on a completely analytic view of the manifolds, yeah, which is not alone in that, but it's, it reinforces that. I don't see that as an objection. You don't get far in geometry without calculating with the number line. Of course not. Sure, sure. So I know problems with that, if that's what you mean by analytic, but if it has the number line according to and the notion of C-infinity functions, then you are limiting your... That's what I meant. Yeah, that's exactly what I meant. Yeah. I was saying it was too... Yes, yes. Do you think that these problems with clarifying the right notion of open set in the case of differential geometry, in the case of manifolds, are related at all to the deeper problems which, you know, you've identified in basing topology on the notion of... ...open set rather than on some arguably more fundamental notion such as that of figure type or generating figure. I mean all the problems that topologists themselves recognized even back in the 40s and 50s. Yeah. So it's variable-intensive quantities whose naturality should be, according to my life, governed by... ... with respect to incidents in relation to figures. Exactly, yes, yes. Going that way, they're sort of being very important auxiliary to the study of figures. Yeah.

30:00 But you can't define the figures in relation to them. They're basically related to, you know, analysis. Yeah. Certainly, that's what I understood, certainly. It's tied up this whole idea of the co-adacracy and how much can you do with the algebra of intensively variable quantities of any kind, let alone Saprinsky values. Yes, which is just one illustration of the issue. No, it's, yes, but we have to go in this next entrance here. Well, again, it underlines the point that you were making in passing this morning, that traditional narrow sense logic, of course, just falls into place as one corner of this picture, which is in terms of the relationship between intensive and extensive quantity and the supports. Thanks. Which is why I think when talking to people like John and the people in Bristol, who of course are mostly going to be people with a background in logic, it's useful to start with this to provide the reorientation that they need. Yeah, we should be in there. Hi, sorry to keep you guys.

32:30 Nice to meet you. I'm Michael Wright. I'm a... You're a mathematician, aren't you? Not actually, no. I'm... I started... I cut my teeth as a mathematician in Cambridge many, many, many years ago. Then I moved over into philosophy and then into the kind of history and philosophy of math. And these days I'm actually a kind of mathematical archivist. I run an archive in France called the Archive for Mathematical Sciences and Philosophy. I'm actually based in France in... I spent most of my time in Paris, but actually in Brittany and Fougere, a little town, which is where I come from. No, no, I just happen to be alive, happen to have been a friend of Bill's for the last 20 years, and I'm also... In fact, for the same length of time, for the last 20 years, a very close friend of John Mabry's, and of Andre, who just gave the talk, and indeed of James, and of Richard, so I've got kind of personal connections, and indeed of Arisha over there, so I've got kind of personal connections with all these people, the longest standing ones, of course, with Bill and with John, because obviously I'm more their age, but we've known each other for 20 years. In fact, I actually introduced them to each other in Cambridge in 1989.

35:00 When Bill came to give a series of lectures, and then we had a workshop on foundations of mathematics and on the role of category theory and class theory, its kind of impact on thinking about the philosophy of maths and on ideas about foundations, and he gave a series of lectures, and then we were actually published as a couple of special, a double, special double number of Philosophia Mathematica a few years afterwards, two, three years afterwards. So I put that together and really since then I've been working more or less full-time in connection with this archive. The archive is actually called the Archive for Mathematical Sciences and Philosophy. But don't do a Google search for it because at the moment our website has been taken down because it got awful. And it's been completely reconstructed. And we're waiting to hear whether we're going to get a grant. Well actually we just told we are going to get a grant. A small but still indispensable one from the French, well actually not directly from the CNRS, but from an outfit called Larsen, so I won't bother you with stories about French. We're going to be allowed to redesign and put up a really decent website and put about the first thousand of our recordings online. I should say the archive is an archive of recordings of seminars, lectures, conferences, interviews with people, mostly audio. But in the last 20 years, a lot of video as well. Fantastic, that's a very good idea. I wish more people would do that. Well, yeah, but at the moment we've got 35,000 recordings sitting at the moment in our base in Fougere, which go back to the 1970s, well, in fact, long before I was involved with it, they go back to about 1971. The most interesting ones, the ones for instance almost everything that builds since 1989 we've been recording, you can see why I'm sure, but also lots of other people as well, both in physics and in history and philosophy of science as well. It's not just me, there are about six or seven other people who, on an entirely voluntary basis, here in Bristol, in Leeds, in London, in Oxford, in Cambridge, in Paris and in other places, including in the States, record stuff. So we've got this huge collection of about 35,000. These days, of course, it's all digital files, but everything before 2000 is on tape.

37:30 The big problem we have is that we just have enough money to keep. ...just enough to stay afloat, which we get. Our main supporter is a foundation in Sweden, a private foundation in Sweden, and that gives me just enough money to draw a small sailor in to keep the thing afloat. But we haven't until now had anything like the money we would need to start digitizing this huge backlog of recordings and start getting it online. But this year it looks as if we may at last have something which will allow us to start. Well, I mean, I have actually been digitising stuff. I've been digitising whenever I get the spare moment. Exactly. I mean, it's a one-man band at the moment. And you can imagine, just to complete a catalogue of 35,000 recordings is going... And each one must take, well, each take plays for 90 minutes and I'm guessing... Well, yes, the dreadful thing is that with the technology that we've got at the moment... If you want to digitize it, which I don't know if you know about these things, there's a thing called audacity, which is just a box standard. That's all we can afford at the moment. That you can only digitize the material in real time. In fact, it's even worse than that. Most of these tapes were recorded at half speed in order to save tape, so you've got to not only re-record them and obviously adjust the running speed, but you've then got to process the things to alter the frequency and amplitude of the signal so as to get a good effect. The main reason for digitizing them is not so much as to enhance the quality of the listening, although it does tend to improve it, but just to simply conserve them because, you know, after 35, 36 years, you know, they won't be readable anymore, curiously, the earliest ones, the ones which were called in the early 70s. There's no problem with those. What happened was in the early 80s, around 82, the manufacturers of audio tapes started to drop the manufacturing standards very sharply, so they started to see a flood of cheap, you know, Taiwanese imports. So Baz Erf and Phillips and all these other people, who until then had actually made really rather high quality tape with good, durable, thick magnetic coating, started reducing the thickness of the magnetic coating and generally producing much shoddier products. Obviously they didn't tell the world that's what they were doing, but of course they were.

40:00 All that stuff that was recorded from about 82, 83 onwards till about the early 90s, a lot of that has already started to deteriorate badly, one or two of them has already been completely lost. There is a very sophisticated technology which was developed by the National Archives here in the UK to go in conjunction with Cambridge, in conjunction with the sound engineers in Cambridge engineering laboratories, called CEDA, which stands for Computer Enhanced Digital Audio Restoration. And that's absolutely amazing. That will restore signals in decayed analogue media and allow you to, I mean, even stuff which when you listen to it sounds like it's hardly anything but white noise, it will extract and, I mean, I don't say we can do anything up or miracles, but within the limits of, you know, digital technology. It's fantastic. We would love to acquire a set of that and use it, not only because it would allow us to restore those of the older recordings which have deteriorated a bit, which fortunately at the moment are still not many. But also because it would allow us to cascade up to 32 audio tracks simultaneously through digital sound boards, so we'd be able to digitise the whole thing probably in a bit of a, well, not very quickly, 35,000 recordings, still going to take probably two or three years, even with a full-time assistant, but it would be a viable project. It's something which could be done within a realistic budget within two to three years. And then you'd have to be able to put the whole lot on the line. The trouble is that that software costs £20,000. It's still essentially a professional forensic sound lab tool. It's the thing which... It's the thing which the spooks and the police use when they're listening for Bin Laden, you know, that kind of thing, it's very, obviously it's commercially available but only very expensive even now, it has begun to come down a bit but it's a long time before, it's not the kind of thing you're going to be, you know, did you download for free on the web, so 20,000 quid, and no, no open source. Well, there are open source alternatives to Audacity, which are a bit better, but they take a bit of time to learn, and you've still got to pay for the hardware anyway, which of course is the major, because even, I mean, that's, actually the hardware for Cedar is not the problem, the hardware is only perhaps three or four thousand quid, you know, you've got sort of a set of, you know, a load of digital soundboards and stuff like that, but the expense is actually the software. But, you know, we're working on the problem, if some, you know...

42:30 There's some retro benefactor that comes along and gives us something we could really start doing, but as it is, we'll get enough this year to start putting stuff online. I hope by the end of this year that we might have 600 or 1,000 of the, what we call the Crown Jewels, the most interesting reforms online, plus the whole catalogue and a really decent site. And so, you know, that's what I'm devoting my life to at the moment. That sounds fascinating. I'm definitely looking forward to seeing it. Well, it's very nice to... When will it be, I mean, is it going to be publicly available? Oh, yes, yes, it'll be in open access. Heavens, yes. Oh, no, we haven't got any commercial... We've always intended to make it a 100% open access archive. Just let me go and introduce you. Are you coming today? You bet. Yes, yes, yes. I'm staying down here with Bill Hernandez. That was Anders Kopp, by the way, the other gentleman was talking about. He's a very distinguished category theorist. It's all very exciting for me to be alive. I've been given about the last six weeks to learn computer theory, so I don't have a history of it. Well, I had the same experience when I met Bill with Cusdam. I don't think I ever got there even now, but at least I... And so were there a bit? Oh, yes, of course it's... I mean, even a... well, it would be quite impossible if it was anything else, but you've probably got a pretty good feel for the landscape here. And for the key... I mean, why it is interesting. It's beautifully and really important because it does introduce a completely different conceptual perspective on almost all the issues in foundations. What's the motivation? Convention or foundation? Completely. It's not even talking the same language. Well, it has a different, it has a quite different answer to the question of what is a foundation. It has quite a different conception of what. The very notion of foundations ought to comprise, and one which I would say is much more realistic, much more in touch with mathematics itself. Well, it's practically, it's more practically motivated, but... Not only that, but... This is the thing that we were talking about when we were learning, when we had a group on category theory, and that is that how, what sort of practical utility is category theory in practice, for instance? If you want to take, because it's often said that, you know, you can take a theorem and... One in one domain of mathematics and then use category theory is useful for translating it into a theory for another branch of mathematics. We were just kind of questioning exactly how it's related, because you always need to come up with concrete examples. There are many. I mean, of course, it's used... I mean, category was developed not as an independent, as it were, autonomous discipline,

45:00 in general category, as a foundational system. It emerged that it could provide that, but it was originally developed for very concrete problems in algebraic and algebraic topology. And particularly cohomology theories, which are the core of modern algebraic topology and algebraic geometry, which I think many mathematicians are actually the core of, really, the strategic centre of mathematics in the last 40 or 50 years. And it was developed as machinery for particularly the notion of scheme and indeed the notion of topos, which was originally defined by Grotenbeek. And then, of course, we've taken the transform of this guy here, who showed how it could be used as a natural way of doing logical sets from a categorical standpoint and some other things. But Grotting's original topos construction was intended basically as a... Machinery for defining what are technically there as sites, sites and coverings, to... He derived long cohomology sequences of the kind that he needed to tackle the vague conjectures, which were the toughest, which were the kind of ever-resting, and still barked into the greatest single intellectual achievement of mathematics in the second half of the 20th century, apart from Freudian phallus theorem, which also was done almost entirely because of the machinery that Grosvenor provided. A great deal of these fundamental notions, like the recognition of just how important and fundamental the notion of functor was, particularly of algebraic functor, that, as it were, all fell into place along the way in the course of what were very definitely concretely motivated mathematical research programmes. So it wasn't, it never was what Steenwold called it in a general abstract sense. I mean it's curious because Steenwold actually contributed a good deal to it and was very very very favorably disposed towards it and he just had a witty turn of phrase but unfortunately that particular one has stuck in people's memories even though he intended it as a kind of rather affectionate leg-pulling remark but But of course it was never general abstract nonsense. It was motivated by very concrete problems to do with schemes, to do with sequences in learning, and to do with problems, in fact the original problem which led Maclean and Heilenberg to define the notion of category in the first place in their 1942-45 papers.

47:30 They wrote three papers, two appeared in the wall, one listed at the end of the wall. The general theory of natural equivalences is the famous one, in which they actually introduce the axioms for the category and for the notion of natural equivalence. But those were motivated by absolutely concrete problems. In fact, they were to do with trying to solve a very concrete problem to do... With the solenoid, which is a particular, the topological space is, yeah, absolutely, they were to do with extremely concrete, it all came out of, well, I say concrete, it's relative, obviously, to any branch. Can you just dig a little bit about that, exactly what was the problem with the solenoid? It was to do with the transformation of, it was to do with, basically, with the homology group, the correct homology group of the space. I'm talking now about solenoid as a notion of topology. Topology. Yeah, yeah, sure, sure. They were trying to compute the homology group, the solenoid. And I ought to know a bit more about history here. You know, I am an archivist these days. I mean, Bill will be able to tell you in no time at all. Oh no, actually, I see when I say no time at all. Yes, that's true. You'll be trapped for 25 minutes. But if you come and have dinner, I'm sure we'll hear it anyway. There was a, in fact, can you remember, I'm still trying to catch your name. James. James here was asking me about the origin, the original paper by McLean and Eilenberg on the general theory of natural equivalencies. Yeah. And, as I recall, the context. They were working, and obviously they had done all this work in homology, homological algebra, and they were specifically trying to compute the homology.

50:00 The point I was trying to focus on was that it was a very specific and concrete problem in a highly developed branch of algebraic topology. It was never inspired by any... I think that began to fall into place after 1958, after the discovery of the adjoint functor theorem and the unidal element. That was when I think that people started to see it as a general framework for the re-description of structure throughout. Not just as an extremely valuable tool involving obviously, you know, a great deal of powerful generalization of concepts, but essentially a very powerful piece of machinery in essentially in algebraic topology and of course increasingly because of growth in functional analysis and then in algebraic geometry. But actually seeing it as a tool for the re-description of structure throughout the whole of mathematics. In terms of concepts which had come out of homology but which in fact provided a framework for I think that really only started to come into focus after, say, about 1958, if you look at the history. Well, Adjoin and Billsworth. Yeah, and Billsworth was a very crucial point of that, part of that. That was absolutely... I think the ubiquity and the importance of Adjoin. Yeah, yeah, yes, yes. Well, he's really the man who made people aware of, precisely at that point, of the ubiquity and importance of adjoin, of adjoin relationships between funders. As just being everywhere in mathematics and providing the key to a new way of conceiving of how one could unify structure across many different mathematical fields and of course hence provide a different... ...perspective on the question of what one should count as a foundation of mathematics, in what sense one should even think of foundations. I mean all of that really came out of his work and also of Kahn's. Because conventional foundations of mathematics is convoluted quite a bit. In mathematics, you want to prove things, you want to convince people beyond all possible doubt. Security, yeah.

52:30 It's true. And that's a strong reason for taking set theory to be a foundation. Sure, because it appears to provide the absolute, the ultimate ingredient, the definition for all further... Well, it sort of says, if you believe this, then... Yes, yes, yes. ...believe all these other constructions. I mean, it provides, as it were, it's providing axioms in the old-fashioned pre-Hilbertian sense of axiom, the way the Greeks thought of axioms. As you know, as ultimate truths about some subject matter that is somehow, which at least in some, obviously, versions of set theoretical equations, some versions. Perhaps not so in fashion these days, it's also supposed to provide a kind of general unifying ontology for mathematics as well. Well, you know, obviously category theory is absolutely not about that. It's this kind of almost... But then the question really pops up, why should I believe on mathematics? That's what the set of theoretical foundations is going to say. I just need to explain what you're talking about. Well, that's what we're here to do. That's the reason for this week. That's exactly why we're here. And that's partly a sociological problem. That's partly the old fallacy of sunken costs problem. People have invested an awful lot of time and energy and effort in mastering, you know, the cumulative hierarchy, not mastering what I think of as now as the sort of mid-century. But the problem we see here as a foundation is, I think it's two-fold. One, I think it's just conception. I misconceived from beginning to end. Two, because I think it... I think that as far as foundations are concerned, there are at least three... There are quite distinct dimensions in which any body of concepts acts as a foundation for, not just for mathematics, but for any other field. One has to do with ontology, one has to do with epistemology, and the other has to do with logic and semantics, with providing theory of meaning. And set theory, of course, claims to do that at least in the first and the third sense for mathematics. The second, the third claim is really shot to ribbons now, and virtually since Godel, but we won't go there. I don't think mathematics has anything like a unified semantics, or if it does, it's something we haven't yet conceived. But the big problem is that...

55:00 Set theory just doesn't bring together these three, at least three distinct dimensions of foundations, whereas category theory does bring them together in a very convincing way. And the other problem that I see is that set theory is certainly axiomatic set theory, the kind of set theory that deals with what people who are professionally aware of them are able to answer. There will be people who say, well, so what? Because why should you assume that a foundation should actually... You know, you're confusing foundations and organisation, and they're two quite different issues. Yeah, well, that's the issue. It does not need to be a practical issue. That was essentially... There have been people like Kreisel and Pfeffermann who have said that, you know, this, you are confusing the conceptual organisation of mathematics, having a kind of unifying framework, which is indeed a very important project, very interesting, exciting, blah, blah, blah. But is in no sense foundational, with foundations which are to do with what we've traditionally always said they were to do with, which is to do with the absolute ultimate epistemological security of your starting points, et cetera, what the kind of Greeks thought of as the requirement of a, you know, without the very notion of axiom came into kind of focus for the first time. And that shouldn't, that doesn't necessarily have anything to do with issues of organization and I mean I can understand that answer I just think it's wrong. I think that the more you go into what category theory does provide, particularly the centrality at this point about structure-preserving mappings, and indeed the very notion of isomorphism is actually a restricted case of this much more general notion of mapping, which is absolutely fundamental and indispensable, which is why I say that I think category theory does. ...provide the key to unifying the ontological, physical, logical, logical, semantic ingredients of the notion of foundation in a way that set theory doesn't, I think you become more and more, well, the more and more. You go into it, the more I'm convinced that category theory does actually have a good answer to those questions.

57:30 The idea isn't, as a foundationalist, to find an axiomatic foundation. No, no, no, absolutely not. No, I agree. I'm with John on that. I mean, you've probably heard. No, I'm with John on that. No, in fact, it's great confusion. Great confusion's been sown by that. Great confusion's been sown by that, because the whole point is, where does the semantics of the axiomatic set that come from in the first place? In order to serve as a foundation... What does a category theorist mean by foundations? Do they mean by pointing out structures that all mathematicians are interested in? That serve to unify all of mathematics. To me that looks like a top-down view. Yes, it is a top-down view. This is one of the differences of perspective. Undoubtedly, category theory does take a more top-down than bottom-up view of mathematics. I mean, I think they would say that in the long run they're going to show what the correct bottom-up view is by... It is much more top-down. I think that's partly because its inspiration is far more from net geometry than from arithmetic, and because it does deal well... I mean, there's also this even deeper conceptual issue of which is the more fundamental, which is the more fundamental both ontologically and epistemologically, our notion of space or our notion of number. Which seems to me to be the kind of issue that certainly is at the core of any philosophy of mathematics, that certainly is a big issue for anybody who's interested as a philosopher in mathematics, and that is something that I think theory does address. So, yeah, and of course the story there is that, you know, one that has to be told through the history of the last, say, 150 years of mathematics, which is... Along came these guys in the 19th century, like Neil Strauss and Dodecinde, in the path of, actually it has to be said, of the kind of, okay, there was the incredible... There was an incredible sort of deepening of geometry by Riemer and his successors, but at the same time that deepening, because of the technical tools it relied on, seemed to make the idea of geometry as actually providing some ultimate foundation, either ontological or epistemological, for mathematics more and more unavailable, and people rejected the Kantian notion. Clearly, geometry was no longer the theory of the structure of physical space, it became analytic, well that was part of it, that was one part of this development, but also because of, in analysis, all these weird, these pathological functions, like the nowhere.

1:00:00 You know, the everywhere continuous but nowhere differentiable curve, the Jordan curve theorem, and the Sierpinski-Sonova thing, which are obviously completely counterintuitive and unphysical and completely unlike anything which the 18th century would have recognized as having to do with the subject matter of geometry. The foundations had to be reconstructed to accommodate these things, or so it appeared. They had to be reconstructed. I think there's quite other stories to be told about that, but I won't go there now. And they were reconstructed in a way which turned on extending the notion of function. And so that's the direction that the arithmetization of analysis took, and with it what became Well, it certainly became put across in sort of philosophy, maths, and indeed undergraduate courses. Math itself has foundations, which was essentially set theory, though you're quite right, I mean, it's a method. So, Frankl's axiomatic theory can't, for the reasons I think John has quite convinced me of, can't be the foundation. No axiomatic theory could be, because you need naive set theory in order to have the semantics for that theory. And the other argument, which of course is that you can't, don't have, you don't really have an ungrounded, you don't have... You don't have a properly grounded notion of structure in the first place, unless you've got some ultimate ingredient of definition of structural notions, which rigorously can only be provided by set theory. That's the claim that John... Did you try to say it wasn't kidney failure? No, it wasn't kidney failure. It wasn't bad. It was meat and potatoes. You missed a great, great, great seminar. I'm so sorry.

1:02:30 You can listen to it all, I got it all, and indeed the workshop talks as well, but that was exceptionally good. Ereshia, I haven't had a chance to say hello, how are you? You look so lovely. Thank you. We are French now. Did you have your hair done by yourself? You do look lovely. It looks really nice. You didn't have your hair done by yourself. This is just accidental. Well, you look really stunning. Did you get everything sorted out in London? So where are you in Paris, André? So, I'm going to Paris, aren't I? When are you going? To Paris. Yes, but when? For the 14th. On the 14th, the 14th of April, okay. Oh, so you're going to be here for the whole of this workshop, that's great. Had you met Bill before now? No, no. So, have you been properly introduced? Yes, but not at the end of this workshop, I'm actually going to be staying around for I'm going to go up to Scotland to record the meeting. They're having a meeting in Edinburgh for the 80th birthday of Michael Atiyah and I've got, just literally in the last three days, I've just been told I've got a tiny little grudge, only 150 quid, but it's just enough to allow me to go up there and record what I'd like to do. We're just waiting for Bill to finish in the piece foire and then we're going to enter. Make a movie. Oh, Christ, yes, that would have been it. I thought you were saying don't go and... Well, we could empty the building pretty fast. Yeah, that's certainly true. They expected it was some high-spirited undergraduate. Yes, yes, that's right. Where it was. Back from half term. And it was, and he was trapped in the body of a 58-year-old. I said he had a long and agonizing discussion with my family. Oh. A. So about 7.30, which was absolutely hard. In fact, yesterday was a lot harder, I thought, I guess because...

1:10:00 The first lecture was firing on all cylinders all morning and then Matthias, the research student, spoke in the afternoon and it was really rather technical, well for me it was very technical. And of course, theoretic stuff. But today we got back more to broader brush, conceptual issues and philosophical issues. Well, I've recorded everything. That's what I've actually been paid by the British Academy to do. It's not the same as being there, but you can still listen to it all. I don't know, I'm sure you always shout at the television set when everybody says anything dumb, so I don't see why, and in fact it's all easy because you can just simply stop the thing, you can stop the recording and shout as much as you like. Yeah, but nobody hears me. Yeah. No, we are starting. We're continuing at 10 tomorrow. That was the agreed plan, wasn't it? That's right. Yeah, that's right, yeah. But we're not going to, we're only going to go on for about, until about 12.30 because Bill's obviously got to get ready to talk. He's talking at 2.30, so we're going to do about two, maybe two and a half hours tomorrow morning. Oh really, is that going to be in the seminar as well? Yeah, yeah, yeah, yeah. What's the topic of discussion? We have actually, we met for a planning meeting yesterday morning, but no planning ever got done. We just launched straight into a freewheeling discussion, which involved Bill giving the equivalent of not four, but probably... There were 12 talks in the course of the two days. And Mateus, the Argentine guy, who gave a very, very interesting expo. And then today, Andre. And Anders, of course. So I don't think we haven't, I think it might not be a bad idea, I was going to suggest to Richard, if we did have a little bit more of a, just a slightly structured agenda, deciding which topics we're going to cover, but I'll wait till the right moment for raising that.

1:12:30 But Bill wanted it to be as informal and freewheeling as possible. They do seem to be a bit stuck, don't they? I wonder if they just forgot them. Maybe we'll try and catch their eye. Yes, just what we were talking about today, of course. James goes to Dubrovnik. Yes, I heard him say. I didn't even realize. I know you did. I know you did. I know you did. But no, I thought the Dubrovnik thing had already taken place. When does it? No, I think it's next week. It's next week, is it? And this year can be interesting. I saw they're going to think metaphysics and science. I saw they had three or four topics. Lots of physics. Like this year. Yes, yes, I know. I looked at the programme, it looked very interesting. But alas, I can't be in three places at once. I think I do quite well managing to be in two, two and a half. Is James going to be talking there, do you know? I think so. In Dubrovnik, everybody is. It's a really nice company. I would like to have gone there. If you have any chance. I've always wanted just to go to Dubrovnik, it's such a beautiful place. We'll plan everything in our class. Proofs are closed under products. You've got to actually exhibit the product. I mean, of course you can say it, because everybody knows that you can exhibit the product. But that's because your audience is sophisticated. But strictly speaking, you've got to exhibit the product. Thank you for your attention. Oh, it's not a question of doubting, is it?

1:15:00 Well, in some things, sometimes we're advocating as we used to respond. We wish we can. Yes, well, that's got applications all over, not just in mathematics. True. I mean, I mean, of course, of course, the category group is both of course, but that's because we all learn. Not a construct, but we were kids. Yeah, but that's why in Euclid we have this notion of possible, right, distribution for action, so we have possible examples that we know to construct. The category theoretic definition of motion, of product, tells you what it is, essentially, right? Exactly, it tells you exactly what you need to know to understand what a product is, okay? I don't agree that... I agree it provides the ultimate ingredients of definition of the notion. It tells you just what it is. But I don't think that is the same thing as what you call existence. What's wrong with it? Well, that's what existence is and what everybody else does. There exists... Well, yeah, but what is the meaning of the existential quantifier? There exists. No, and if all you had were relative consistency proofs for all the theories in which you define all these structures, what would you be missing that what you call having an existence proof? I don't define them in some sort of... ...highfalutin theory. I just... No, don't have to be highfalutin. Don't introduce raref rhetoric here. They can be very low-level theories, but if you have a relative consistency proof of the theories in which these things are defined, and the consistency proofs, as it were, all mesh, what is it that you would not have that what you call an existence proof gives you?

1:17:30 That's what I need to know. I'm just saying, I think Anders... Anders, you'll have to come to my rescue. No, I think Anders thinks that's a fair question. Does Anders think it's a fair question or not? It's of course obvious to anybody that knows any algebra that the category of groups is closed under product. That's obvious. But what makes it obvious is that you can construct a particular... Well, you can construct the product. Yes, but use the question. Why is it obvious to be categorical? That's the problem. Well, it's the same argument. It's the same argument. Yes, but I was going this afternoon in a discussion about axiomatic set theory in the sense that we defined it with an order of care defined in the final way. There, Gruber-Kirchweil You take all of them to Pinnacle Mountain, so why is it that all of them in all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, all of them, It's consistent to assume that it exists. And if you think that the middle triangle is consistent, you won't do it that way, if you really want to. Yeah, but I'm talking about the kind of set theory you use when you're teaching an emphasis. I'm not talking about some absolute logical theory. When you're teaching emphasis, the notion of motion is a premise of motion. Yeah. Well, not the way I taught it. I've taught out of Spivak's books, and he gives the... the Kuratowski definition of ordered pair is actually quite natural, because... Well, the Kuratowski... Kuratowski defined a linear ordering to be the set of all of its initial set... Okay, so if you've got a two... if you've got a two-element ordering... Well, I mean, no, it's quite simple.

1:20:00 That's what I think of it. An ordering of, because you've got the elements ABC and you want to take them in the order BCA. Well then, you just take the set consisting of... the set consisting of B, the set consisting of BC, and the set consisting of BCA. So, you just take the set of initial segments, and that's the order. And that turns the ordering into a set. So, I mean... So then an ordered pair, if the two elements of the pair are different, it's just the linear ordering of the operators. And if they're the same, it's just the one element, the linearity of links. I mean, these aren't inevitable, but they're quite plausible ways of thinking. Yeah, I was going to say, why is that any more natural than the definition that Andy just gave? What he didn't give me, what he said, it was primitive. No, I thought he gave a definition involving map spaces. I don't believe you should take things as primitive unless you have to. No, I completely agree. Well, then you don't have to in the case of ordered power. But then I can take the notion of morphism as primitive. In fact, I do. No, because it is a high-level concept. Of course you can say, you can give an axiomatic definition of category and say an arrow is whatever satisfies any interpretation. Thank you for your attention. I posit such and such. Well, what gives you the right to do that? Well, it's essentially Frege's argument against Hilbert, isn't it?

1:22:30 Well, but it's perfectly valid. Why do you grant a license to a mathematician that you wouldn't grant to a theologian? I positively go on. Well, and he's unique. And you say, well, how do you justify that? I don't justify it. That's my policy. Well, you can't do that. I mean, intellectually, it's something funny. Except that I don't think it is at all like positing a god, because positing a god is, because theology is disconnected from the whole of science and the whole of the rest of discourse in a way. Most certainly isn't. Most of the great scientists of the past have been religious believers of one religion or another. All those things are true, but I'm speaking about a person who's a Christian. I mean, the difference between binary products and the first possible product by the universal property and the God is that their binary product is not unique but only unique by the method. You can't say that about God. That doesn't work. Yeah, that certainly doesn't work, exactly. But they all say exactly the same thing. Well, I don't know. I mean, look, I mean, the category, the concrete category of the group is going to get to some groups and some groups more. I mean, that's quite straightforward. It's closed into products. How do you know? Because, well, this construction that we used to make, because when you teach group theory, you teach it that way. And you say that the product is done pairwise and all of a sudden... Well, I mean, we're confusing the issue by talking about parts and groups. Ah, the general notion of product... No, no, I mean, from this viewpoint, it's not the same. It's quite clear that the problem is introduced by a problem of two steps. The foundation of this is one problem of two steps.

1:25:00 Yes, because the product of two groups, you take the underlying sets of both, you take the set theoretic of both. It's truly trivial that this underlying function resolves binary products. So the foundational issue, what is the binary product of two sets, does not exist. And Kruber-Keefe made the bold step of taking a definitive notion in contrast to the other. This goes for my math, and that goes for your math. It's a poor job. It's a poor job. But why posit something when you can define it? Well, it's because of the... Because you're going to have to define it from other things which you take as primitive, and you may be less happy about taking those as primitive. For example, if you're doing conventional differential geometry, you can define... A metaphor is something that has, and actually has... You can include in the definition the existence that actually has at each point a tangent space. A copy of Rn, so. You can put that into the definition. People don't because you can construct the tangent space from stuff that's already there. I mean, actually, when you're teaching it, that's the best way to teach it. You can teach in different ways, John. I don't understand. But you can teach in different ways, just saying group is a category without an object. It's also intuitively appealing, but in a different way. There are other alternatives. I mean, there are more spectacular ones with topological spaces, because you can define topological spaces in terms of their open space. All of these terms are equivalent.

1:27:30 Well, I mean, it's easy to see that a categorical group and a conventional group, that they're completely interchanged. Yeah. It's easy to see how you define the category, and it's easy to see, given the category, how you define the associated group, and it's a, you know... But to be precise, you would need to have, say, kind of, justify equivalence to make kind of larger category and just say, okay, there is this translation, there is backward, and they cancel each other to ideal, both sides, something like this. But you're going to, I mean, you're looking at... You're trying to look at the thing more generally, which of course is the way you should look at it. That's the beauty of it. You have abstracted away from the fact that you're dealing with groups or with sets or whatever, you've just got the machinery, the minimal machinery of a category, and now you give a perfectly rigorous definition of what it is to be a product in that category. Thank you for your attention. I don't think I don't think he's right about it I think they're all the origin of the notion of ordered pairs clearly in syntax ordered pairs are labeled we actually write down But isn't that also true of initial segment, which is what Torotovsky bases his definition on, right? I mean, how do you define what an initial segment is if it's not syntactic? Well, because it's a certain set. Sorry.

1:30:00 Under some sort of ordering. I mean, you could say it. You don't have to say it the way I said it. In fact, you have to say it in a more complicated way to avoid starting with the ordering already. I'm just saying, by example, if you want to take the set, if you just want to take A, B, and C and impose the order in which, say, linear ordering in which B comes first, A comes second, and C comes third. You just take the set. There's got three members. One of them consists of B alone, then singleton, then doubleton BA, and then BAC. And that completely and unambiguously says what a linear... Yeah, but it just depends on the hypothesis that non-ordinary pair is somehow primary. And it can be supported with some kind of intuition. But you also can make a case just saying ordered pair is primary. Why? What makes you say you can do that? You can't do it when you're talking about bosons because you're talking. No, no, but talking about simple things. It may be closer to the way we think because we think in terms of pictures. A linear ordering, a three-element linear ordering is obtained by putting little square brackets and writing down A, B, C. And reading it from left to right. That is to incorporate all these, this mixture of syntactics and so on into a purely, what ought to be a purely logical notion.

1:32:30 Well, here I would appeal to history. A dangerous thing for a man who's just said that things must be made absolutely logical and aseptic. You can't appeal to syntax, let alone history. There's nothing wrong with doing it that way. That's what happened. But what I'm saying is that historically the notion of a finite collection came first. Partly because people didn't use algebra to write things down. Still before set theory it was not absolute. Descartes realized you could label every point by a pair of numbers. No, if I just report the natural numbers, I would say that's an example that one thing comes first and the other one comes next.